Abstract
The authors provide a survey of recent results in special functions of classical analysis and geometric function theory, in particular, the circular and hyperbolic functions, the gamma function, the elliptic integrals, the Gaussian hypergeometric function, power series, and mean values.
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1 Introduction
The study of quasiconformal maps led the first two authors in their joint work with Vamanamurthy to formulate open problems or questions involving special functions [14, 16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.
The present paper is the third in a series of surveys by the first two authors, the previous papers [20, 23] being written jointly with the late Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14, 16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors acknowledge his crucial role in our joint work. For an update to the bibliographies of [20, 23] the reader is referred to [12].
In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l’Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [21, Theorem 2].
Theorem 1.1 (l’Hôpital Monotone Rule).
Let −∞ < a < b < ∞, and let \(f,g: [a,b] \rightarrow \mathbb{R}\) be continuous functions that are differentiable on (a,b), with \(f(a) = g(a) = 0\) or \(f(b) = g(b) = 0.\) Assume that g′(x) ≠ 0 for each x ∈ (a,b). If f′∕g′ is increasing (decreasing) on (a,b), then so is f∕g.
Theorem 1.1 assumes that a and b are finite, but the rule can be extended easily by similar methods to the case where a or b is infinite. The LMR has been used effectively in the study of the monotonicity of a quotient of two functions. For instance, Pinelis’ note [146] shows the potential of the LMR. As a complement to Pinelis’ note, the paper [21] contains many applications of LMR in calculus. Also the history of LMR is reviewed there.
In this survey we give an account of the work in the special functions of classical analysis and geometric function theory since our second survey. In many of these results the LMR was an essential tool. Because of practical constraints, we have had to exclude many fine papers and have limited our bibliography to those papers most closely connected to our work.
The aim of our work on special functions has been to solve open problems in quasiconformal mapping theory. In particular, we tried to settle Mori’s conjecture for quasiconformal mappings [127] (see also [118, p. 68]). For the formulation of this problem, let K > 1 be fixed and let M(K) be the least constant such that
for every K-quasiconformal mapping f: B → B of the unit disk B onto itself with f(0) = 0. A. Mori conjectured in 1956 that \(M(K) \leqslant 1{6}^{1-1/K}\,.\) This conjecture is still open in 2013. Some of the open problems that we found will be discussed in the last section.
2 Generalizations of Jordan’s Inequality
The LMR application list, begun in [21], led to the Master’s thesis of Visuri, on which [109] is based. Furthermore, applications of LMR to trigonometric inequalities were given in [109]. Numerous further applications to trigonometric functions were found by many authors, and some of these papers are reviewed in this section and the next.
By elementary geometric methods one can prove that
a result known as Jordan’s inequality. In a recent work, Klén et al. [109] have obtained the inequalities
and
Inspired by these results, Lv, Wang, and Chu [121] proved that, for \(a = (\log (\pi /2))/\log \sqrt{2} \approx 1.30299\),
where 4∕3 and a are best constants and that for \(b = (\log \mathrm{sinh}1)/(\log \mathrm{cosh}1) \approx 0.372168,\)
where 1∕3 and b are best constants.
Many authors have generalized or sharpened Jordan’s inequality, either by replacing the bounds by finite series or hyperbolic functions or by obtaining analogous results for other functions such as hyperbolic or Bessel functions. The comprehensive survey paper by Qi et al. [150] gives a clear picture of these developments as of 2009. For example, in 2008 Niu et al. [143] obtained the sharp inequality
for each natural number n, with best possible constants α k and β k . That same year Wu and Srivastava [198] obtained upper and lower estimates on (0, π∕2] for (sinx)∕x that are finite series in powers of (x −θ), where θ ∈ [x, π∕2], while Zhu [211] obtained bounds as finite series in powers of (π 2 − 4x 2). Zhu [210] obtained bounds for (sinx)∕x as finite series in powers of (r 2 − x 2) for 0 < x \( \leqslant \) r \( \leqslant \) π∕2, yielding a new infinite series
Yang [199] showed that a function f admits an infinite series expansion of the above type if and only if f is analytic and even.
In 2011 Huo et al. [97] obtained the following generalization of Jordan’s inequalities:
for t \( \geqslant \) 2, \(n \in \mathbb{N}\), and 0 < x ≤ θ < π, where the coefficients μ k and ω k are defined recursively and are best possible.
More recently, in 2012, Chen and Debnath [74] have proved that, for 0 < x \( \leqslant \) π∕2,
where
and
for any θ \( \geqslant\) 2, with equality when \(x =\pi /2\).
In a recent work Sándor [164] (see also [165, p. 9]) proved that \(h(x) \equiv [\log (x/\sin x)]/\log ((\sinh x)/x)\) is strictly increasing on (0, π∕2). He used this result to prove that the best positive constants p and q for which
is true are p = 1 and \(q = [\log (\pi /2)]/\log ((\sinh (\pi /2))/(\pi /2)) \approx 1.18.\)
In an unpublished manuscript, Barbu and Pişcoran [28] have proved, in particular, that
Kuo [116] has developed a method of obtaining an increasing sequence of lower bounds and a decreasing sequence of upper bounds for (sinx)∕x, and he has conjectured that the two sequences converge uniformly to (sinx)∕x.
Since there is a close connection between the function (sinx)∕x and the Bessel function J 1∕2(x) (cf. [219]), it is natural for authors to seek analogs of the Jordan inequality for Bessel and closely related functions. Baricz and Wu [35, 40], Zhu [219, 220], and Niu et al. [144] have produced inequalities of this type. Zhu [221] has also obtained Jordan-type inequalities for ((sinx)∕x)p for any p > 0. Wu and Debnath [195] have generalized Jordan’s inequality to functions f(x)∕x on [0, θ] such that f is (n + 1)-times differentiable, f(0) = 0, and either n is a positive even integer with f (n+1) increasing on [0, θ] or n is a positive odd integer with f (n+1) decreasing on [0, θ].
3 Other Inequalities Involving Circular and Hyperbolic Functions
3.1 Redheffer
In 1968 Redheffer [157] proposed the problem of showing that
or, equivalently, that
A solution of this problem was provided by Williams [192], using infinite products, who also proved the stronger inequality
Later, using Erdös-Turán series and harmonic analysis, Li and Li [120] proved the double inequality
They also found a method for obtaining new bounds from old for (sinx)∕x, but Kuo [116] gave an example to show that the new bounds are not necessarily stronger. In 2003 Chen et al. [76], using mathematical induction and infinite products, found analogs of the Redheffer inequality for cosx:
and for hyperbolic functions
In 2008, inspired by the inequalities above, Zhu and Sun [224] proved that
with best possible constants α = 1 and \(\beta {=\pi }^{2}/16\), and
with best possible constants γ = 1 and \(\delta {=\pi }^{2}/12\). They obtained similar results for the hyperbolic sine and cosine functions. In 2009 Zhu [216] showed that
holds if and only if α \( \geqslant \) π 2∕6 and β \( \leqslant \) 1, with analogous results for cosx and (tanx)∕x. In 2009 Baricz and Wu [41] and in 2011 Zhu [222] proved Redheffer-type inequalities for Bessel functions.
3.2 Cusa-Huygens
The inequality
was discovered by N. de Cusa in the fifteenth century (cf. [71]) and proved rigorously by Huygens [98] in the seventeenth century. In 2009 Zhu [218] obtained the following inequalities of Cusa-Huygens type:
and
That same year Zhu [214] discovered a more general set of inequalities of Cusa type, from which many other types of inequalities for circular functions can be derived. He proved the following: Let 0 < x < π∕2. If p \( \geqslant \) 1, then
if and only if β \( \leqslant \) 1∕3 and \(\alpha \geqslant 1 - {(2/\pi )}^{p}.\) If 0 \( \leqslant \) p \( \leqslant \) 4∕5, then (3) holds if and only if α \( \geqslant \) 1∕3 and \(\beta \leqslant 1 - {(2/\pi )}^{p}.\) If p < 0, then the second inequality in (3) holds if and only if β \( \geqslant \) 1∕3. In a later paper [219] Zhu obtained estimates for (sinx)∕x and (sinhx)∕x that led to new infinite series for these functions. For some similar results see also [194].
In 2011 Chen and Cheung [71] obtained the sharp Cusa-Huygens-type inequality
for 0 < x < π∕2, with best possible constants \(\alpha = (\log (\pi /2))/\log (3/2) \approx 1.11\) and β = 1.
In 2011 Neuman and Sándor [142] discovered a pair of optimal inequalities for hyperbolic and trigonometric functions, proving that, for 0 < x < π∕2, the best positive constants p and q in the inequality
are \(p = (\log (\pi /2))/\log \cosh (\pi /2) \approx 0.49\) and \(q = 1/3\) and that for x ≠ 0 the best positive constants p and q in the inequality
are \(p = 3/2\) and \(q = (\log 2)/\log [(\sinh (\pi /2))/(\pi /2)] \approx 1.82.\)
3.3 Becker-Stark
In 1978 Becker and Stark [49] obtained the double inequality
where the numerator constants 8 and π 2 are best possible.
In 2008 Zhu and Sun [224] showed that
holds if and only if α \( \leqslant \) π 2∕24 and β \( \geqslant \) 1.
In 2010 Zhu and Hua [223] sharpened the Becker-Stark inequality by proving that
where \(\alpha = 4(8 {-\pi }^{2}){/\pi }^{2} \approx -0.76\) and \(\beta {=\pi }^{2}/3 - 4 \approx -0.71\) are the best possible constants. They also developed a systematic method for obtaining a sequence of sharp inequalities of this sort.
In 2011 Ge [88] obtained
for 0 < x < π∕2. That same year Chen and Cheung [71] proved the sharp Becker-Stark-type inequality
with best possible constants \(\alpha {=\pi }^{2}/12 \approx 0.82\) and β = 1.
3.4 Wilker
In 1989 Wilker [190] posed the problem of proving that
and of finding
Anglesio et al. [191] showed that the function in (5) is decreasing on (0, π∕2), that the value of c in (5) is 16∕π 4, and that, moreover, the supremum of the expression in (5) on (0, π∕2) is 8∕45. Hence
for 0 < x < π∕2, where 16∕π 4 ≈ 0. 164 and 8∕45 ≈ 0. 178 are best possible constants. (Note: [21] erroneously quoted [191] as saying that the function in (5) is increasing.) In 2007 Wu and Srivastava [197] proved the Wilker-type inequality
However, Baricz and Sándor [39] discovered that (7) is implied by (4).
In 2009 Zhu [218] generalized (4) and obtained analogs for hyperbolic functions, showing that, for 0 < x < π∕2, α \( \geqslant \) 1,
and that, for x > 0, α \( \geqslant \) 1,
These two results of Zhu are special cases of a recent lemma due to Neuman [138, Lemma 2].
In 2012 Sándor [162] has proved that, for 0 < x ≤ π∕2, α > 0,
Using power series, Chen and Cheung [72] obtained the following sharper versions of (6):
and
The constants 16∕315 ≈ 0. 051 and (2∕π)6 ≈ 0. 067 in (8) and 104∕4725 ≈ 0. 022 and (2∕π)8 ≈ 0. 027 in (9) are best possible. For 0 < x < π∕2, Chen and Cheung also obtained upper estimates complementary to (7):
and
where the constants 2∕45 and 8∕945 are best possible.
In 2012, Sándor [164] has shown that
and
where \(q = [\log (\pi /2)]/\log [(\sinh (\pi /2))/(\pi /2)] \approx 1.18.\)
Extensions of the generalized Wilker inequality for Bessel functions were obtained by Baricz and Sándor [39] in 2008.
3.5 Huygens
An older inequality due to Huygens [98] is similar in form to (4):
and actually implies (4) (see [141]). In 2009, Zhu [217] obtained the following inequalities of Huygens type:
for all x ∈ (0, π∕2) if and only if p \( \geqslant \) 1∕3 and q \( \leqslant \) 0;
for all x ∈ (0, ∞) if and only if p \( \leqslant \) 1∕3 and q \( \geqslant \) 1;
for all x ∈ (0, π∕2) if and only if p \( \leqslant \) 1∕3 and \(q \geqslant 1 - 2/\pi\); and
for all x ∈ (0, ∞) if and only if p \( \geqslant \) 1∕3 and q \( \leqslant \) 0.
In 2012 Sándor [162] has showed that, for 0 < x ≤ π∕2, α > 0,
Chen and Cheung [72] also found sharper versions of (10) as follows: For 0 < x < π∕2,
and
where the constants \(3/20 = 0.15\) and (2∕π)4 ≈ 0. 16 in (11) and 3∕56 ≈ 0. 054 and (2∕π)6 ≈ 0. 067 in (12) are best possible.
Recently Hua [96] have proved the following sharp inequalities: For 0 < | x | < π∕2,
where the constants 1∕40 and \((80 - 24\pi ){/\pi }^{4}\) are best possible, and, for x ≠ 0,
where the constant 3∕20 is best possible.
3.6 Shafer
The problem of proving
was proposed by Shafer [166] in 1966. Solutions were obtained by Grinstein, Marsh, and Konhauser [169] in 1967. In 2011 Chen, Cheung, and Wang [73] found, for each a > 0, the largest number b and the smallest number c such that the inequalities
are valid for all x \( \geqslant \) 0. Their answer to this question is indicated in the following table:
In 1974, in a numerical analytical context [167], Shafer presented the inequality
which he later proved analytically [168]. In [213] Zhu proved that the constant 80∕3 in Shafer’s inequality is best possible and also obtained the complementary inequality
where 256∕π 2 is the best possible constant.
3.7 Fink
In [132, p. 247], there is a lower bound for arcsinx on [0, 1] that is similar to Shafer’s for arctanx. In 1995 Fink [87] supplied a complementary upper bound. The resulting double inequality is
and both numerator constants are best possible. Further refinements of these inequalities, along with analogous ones for arcsinh x, were obtained by Zhu [212] and by Pan with Zhu [145]. We note that, for 0 < x < 1, this double inequality is equivalent to
in which the second relation is the Cusa inequality.
3.8 Carlson
In 1970 Carlson [67, (1.14)] proved the inequality
In 2012, seeking to sharpen and generalize (14), Chen and Mortici [75] determined, for each fixed c > 0, the largest number a and smallest number b such that the double inequality
is valid for all x ∈ [0, 1]. Their answer to this question is indicated in the following table:
These authors also proved that, for all x ∈ [0, 1], the inequalities
hold on [0, 1], with best constants \(a = (2\root{3}\of{4}-\pi )/\pi \approx 0.01\) and b = 0.
Moreover, in view of the right side of (14), in 2011 Chen, Cheung, and Wang [73] considered functions of the form
on [0, 1] and determined the values of p, q, r such that f(x) is the best third-order approximation of arccosx in a neighborhood of the origin. The answer is that, for \(p = (\pi +2){/\pi }^{2}\), \(q = (\pi -2){/\pi }^{2}\), \(r =\pi /2\), one has
With the values of p, q, r stated above, the authors were led to a new lower bound for arccos:
3.9 Lazarević
In [117] Lazarević proved that, for x ≠ 0,
if and only if q \( \geqslant \) 3. Zhu improved upon this inequality in [215] by showing that if p > 1 or p \( \leqslant \) 8∕15, then
for all x > 0 if and only if q \( \geqslant \) 3(1 − p). For some similar results see also [194].
In 2008 Baricz [34] extended the Lazarević inequality to modified Bessel functions and also deduced some Turán- and Lazarević-type inequalities for the confluent hypergeometric functions.
3.10 Neuman
Neuman [137] has recently established several inequalities involving new combinations of circular and hyperbolic functions. In particular, he has proved that if x ≠ 0, then
and
4 Euler’s Gamma Function
For Rez > 0 the gamma function is defined by
and the definition is extended by analytic continuation to the entire complex plane minus the set of nonpositive integers. This function, discovered by Leonhard Euler in 1729, is a natural generalization of the factorial, because of the functional identity
The gamma function is one of the best-known and most important special functions in mathematics and has been studied intensively.
We begin our treatment of this subject by considering an important special constant discovered by Euler and related to the gamma function.
4.1 The Euler-Mascheroni Constant and Harmonic Numbers
The Euler-Mascheroni constant γ = 0. 5772156649… is defined as
where γ n ≡ H n − logn, \(n \in \mathbb{N}\) and where H n are the harmonic numbers
The number γ is one of the most important constants in mathematics and is useful in analysis, probability theory, number theory, and other branches of pure and applied mathematics. The numerical value of γ is known to 29, 844, 489, 545 decimal places, thanks to computation by Yee and Chan in 2009 [201] (see [77, p. 273]).
The sequence γ n converges very slowly to γ, namely with order 1∕n. By replacing logn in this sequence by \(\log (n + 1/2)\), DeTemple [84] obtained quadratic convergence (see also [69]). In [130] Mortici made a careful study of how convergence is affected by changes in the logarithm term. He introduced new sequences
where P and Q are polynomials with leading coefficient 1 and \(\deg P -\deg Q = 1.\) By judicious choice of the degrees and coefficients of P and Q he was able to produce sequences M n tending to γ with convergence of order 1∕n 4 and 1∕n 6. He also gave a recipe for obtaining sequences converging to γ with order \(1/{n}^{2k+2},\) where k is any positive integer. This study is based on the author’s lemma, proved in [129], that connects the rate of convergence of a convergent sequence {x n } to that of the sequence \(\{x_{n} - x_{n+1}\}\).
In 1997 Negoi [134] showed that if \(T_{n} \equiv H_{n} -\log (n + 1/2 + 1/(24n))\), then \(T_{n} + {[4{n}^{3}]}^{-1}\) is strictly decreasing to γ and \(T_{n} + {[48(n + 1)]}^{-3}\) is strictly increasing to γ, so that \({[48(n + 1)]}^{-3} <\gamma -T_{n} < {[48{n}^{3}]}^{-1}.\) In 2011 Chen [70] established sharper bounds for γ − T n by using a lemma of Mortici [129].
Using another approach, in 2011 Chlebus [77] developed a recursive scheme for modifying the sequence H n − logn to accelerate the convergence to γ to any desired order. The first step in Chlebus’ scheme is equivalent to the DeTemple [84] approximation, while the next step yields a sequence that closely resembles the one due to Negoi [134].
In [8] Alzer studied the harmonic numbers (16), obtaining several new inequalities for them. In particular, for n \( \geqslant \) 2, he proved that
where \(\alpha = (6\sqrt{6} - 2\root{3}\of{396})/(3\log (\log 2+\gamma )) \approx 0.014\) and β = 1 are the best possible constants and γ is the Euler-Mascheroni constant.
4.2 Estimates for the Gamma Function
In [14, Lemma 2.39] Anderson, Vamanamurthy, and Vuorinen proved that
and that the function \((\log \varGamma (1 + x/2))/x\) is strictly increasing from [2, ∞) onto [0, ∞). In [13] Anderson and Qiu showed that \((\log \varGamma (x + 1))/(x\log x)\) is strictly increasing from (1, ∞) onto (1 −γ, 1), where γ is the Euler-Mascheroni constant defined by (15), thereby obtaining the strict inequalities
They also conjectured that the function \((\log \varGamma (x + 1))/(x\log x)\) is concave on (1, ∞), and this conjecture was proved by Elbert and Laforgia in [85, Sect. 3]. One should note that in 1989 Sándor [159] proved that the function \({(\varGamma (x + 1))}^{1/x}\) is strictly concave for x ≥ 7.
Later Alzer [4] was able to extend (18) by proving that, for x ∈ (0, 1),
with best possible constants \(\alpha = 1-\gamma = 0.42278\ldots\) and \(\beta = {(\pi }^{2}/6-\gamma )/2 = 0.53385\ldots\). For x ∈ (1, ∞) Alzer was able to sharpen (18) by showing that (19) holds with best possible constants \(\alpha = {(\pi }^{2}/6-\gamma )/2 \approx 0.534\) and β = 1. His principal new tool was the convolution theorem for Laplace transforms.
Another type of approximation for Γ(x) was derived by Ivády [102] in 2009:
In 2011 Zhao, Guo, and Qi [207] simplified and sharpened (20) by proving that the function
is strictly increasing from (0, 1) onto (γ, 2(1 −γ)), where γ is the Euler-Mascheroni constant. As a consequence, they proved that
with best possible constants \(\alpha = 2(1-\gamma )\) and β = γ.
Very recently Mortici [133] has determined by numerical experiments that the upper estimate in (18) is a better approximation for Γ(x) than the lower one when x is very large. Hence, he has sought estimates of the form Γ(x) ≈ x a(x), where a(x) is close to x − 1 as x approaches infinity. For example, he proves that
where \(a(x) = 1 - 1/\log x + 1/(2x) - (1 - (\log 2\pi )/2)/(x\log x)\) and where \(b(x) = 1 - 1/\log x + 1/(2x)\). The left inequality is valid for x \( \geqslant \) 2. Mortici has also obtained a pair of sharper inequalities of this type, valid for x \( \geqslant \) 2, and has showed how lower and upper estimates of any desired accuracy may be obtained. His proofs are based on an approximation for logΓ(x) in terms of series involving Bernoulli numbers [25, p. 29] and on truncations of an asymptotic series for the function \((\log \varGamma (x))/((x - 1)\log x).\) These results provide improvements of (18).
4.3 Factorials and Stirling’s Formula
The well-known Stirling’s formula for n! ,
discovered by the precocious homeschooled and largely self-taught eighteenth-century Scottish mathematician James Stirling, approximates n! asymptotically in the sense that
Because of the importance of this formula in probability and statistics, number theory, and scientific computations, several authors have sought to replace (21) by a simple sequence that approximates n! more closely (see the discussions in [47, 48]). For example, Burnside [63] proved in 1917 that
that is, \(\lim \limits _{n\rightarrow \infty }(n!/\beta _{n}) = 1.\) In 2008, Batir [47] determined that the best constants a and b such that
are \(a = 1 - 2\pi {e}^{-2} \approx 0.1497\) and \(b = 1/6 \approx 0.1667\). Batir offers a numerical table illustrating that his upper bound formula \({n}^{n+1}{e}^{-n}\sqrt{2\pi }/\sqrt{n - 1/6}\) gives much better approximations to n! than does either (21) or (22).
In a later paper [48] Batir observed that many of the improvements of Stirling’s formula take the form
for some real numbers a and b. Batir sought the pair of constants a and b that would make (24) optimal. He proved that the best pairs (a, b) are (a 1, b 1) and (a 2, b 2), where
and
where \(\lambda = \sqrt{2 + {2}^{2/3 } + {2}^{4/3}} \approx 2.47128\) and a 1 and a 2 are the real roots of the quartic equation \(3{x}^{4} - 4{x}^{3} + {x}^{2} + 1/12 = 0\).
Ramanujan [156] sought to improve Stirling’s formula (21) by replacing \(\sqrt{ 2n}\) in the formula by the sixth root of a cubic polynomial in n:
In this connection there appears in the record also his double inequality, for x \( \geqslant \) 1,
Motivated by this inequality of Ramanujan, the authors of [18] defined the function \(h(x) \equiv u{(x)}^{6} - (8{x}^{3} + 4{x}^{2} + x)\), where \(u(x) = {(e/x)}^{x}\varGamma (x + 1)/\sqrt{\pi },\) and conjectured that h(x) is increasing from (1, ∞) into \((1/100,1/30)\). In 2001 Karatsuba [106] settled this conjecture by showing that h(x) is increasing from [1, ∞) onto [h(1), 1∕30), where \(h(1) = {e}^{6}{/\pi }^{3} - 13 \approx 0.011\).
In an unpublished document, E. A. Karatsuba suggested modifying Ramanujan’s approximation formula (25) by replacing the radical with the 2kth root of a polynomial of degree k and determining the best such asymptotic approximation. Such a program was partially realized by Mortici [132] in 2011, who proposed formula (27) below for k = 4, but the more general problem suggested by Karatsuba remains an open problem. Mortici’s proposed Ramanujan-type asymptotic approximation is as follows:
In connection with (27), he defined the function
where \(u(x) = {(e/x)}^{x}\varGamma (x + 1)/\sqrt{\pi },\) and proved that g(x) is strictly decreasing from [3, ∞) onto (g(∞), g(3)], where \(g(\infty ) = -128/1215 \approx -0.105\) and \(g(3) = 256{e}^{24}/(4304672{1\pi }^{4}) - 218336/135 \approx -0.088.\) Mortici’s method for proving monotonicity was simpler than Karatsuba’s, because he employed an excellent result of Alzer [3] concerning complete monotonicity (see Sect. 4.6 below for definitions). Mortici claimed that his method would also simplify Karatsuba’s proof in [106]. Finally, he proved that, for x \( \geqslant \) 3,
where \(R(x,t) \equiv \root{8}\of{16{x}^{4} + \frac{32} {3} {x}^{3} + \frac{32} {9} {x}^{2} + \frac{176} {405}x - t}\), and \(\alpha = 128/1215\), β = g(3) are the best possible constants.
In 2012 Mahmoud, Alghamdi, and Agarwal [124] deduced a new family of upper bounds for Γ(n + 1) of the form
where ζ is the Hurwitz zeta function
These upper bounds improve Mortici’s inequality (27).
4.4 Volume of the Unit Ball
The volume Ω n of the unit ball in \({\mathbb{R}}^{n}\) is given in terms of the gamma function by the formula
Whereas the volume of the unit cube is 1 in all dimensions, the numbers Ω n strictly increase to the maximum \(\varOmega _{5} = {8\pi }^{2}/15\) and then strictly decrease to 0 as n → ∞ (cf. [60, p.264]). Anderson, Vamanamurthy, and Vuorinen [14] proved that Ω n 1∕n is strictly decreasing and that the series \(\sum _{n=2}^{\infty }\varOmega _{n}^{1/\log n}\) is convergent. In [13] Anderson and Qiu proved that Ω n 1∕(nlogn) is strictly decreasing with limit \({e}^{-1/2}\) as n → ∞.
In 2008 Alzer published a collection of new inequalities for combinations of different dimensions and powers of Ω n [7, Sect. 3]. We quote several of them below:
where the best possible constants are \(a = (4 - 9\pi /8){(2/(\pi e))}^{1/2}/e = 0.0829\ldots\) and \(b {=\pi }^{-1/2} = 0.5641\ldots\);
with best possible constant factors \(a = (4/{e}^{2})(1 - 8/(3\pi )) = 0.0818\ldots\) and \(b = 1/(2\pi ) = 0.1591\ldots\);
with best possible constants \(a = 3\sqrt{2}\pi /(6 + 4\pi ) = 0.7178\ldots\) and \(b = \sqrt{2\pi } = 2.5066\ldots\); and
with best possible constants \(a = (4-\pi )\sqrt{2} = 1.2139\ldots\) and \(b = \sqrt{2\pi }/2 = 1.2533\ldots\).
Alzer’s work in [7] includes a number of new results about the gamma function and its derivatives.
In 2010 Mortici [128], improving on some earlier work of Alzer [5, Theorem 1], obtained, for n \( \geqslant \) 1 on the left and for n \( \geqslant \) 4 on the right,
where \(a = 64 \cdot 72{0}^{11/12} \cdot {2}^{1/22}/(10395 {\cdot \pi }^{5/11}) = 1.5714\ldots.\) He sharpened the work of Alzer [5, Theorem 2] and Qiu and Vuorinen [154] in the following result, valid for n \( \geqslant \) 1:
Mortici also proved, in [128, Theorem 4], that, for n \( \geqslant \) 4,
This result improves a similar one by Alzer [5, Theorem 3, valid for n \( \geqslant \) 1], where the exponent on the left is the constant 2 − log2 π. Very recently, Yin [202] improved Mortici’s result as follows: For n \( \geqslant \) 1,
where \(\beta = {(391/30)}^{1/3}\).
4.5 Digamma and Polygamma Functions
The logarithmic derivative of the gamma function, \(\psi (x) \equiv \frac{d} {dx}\log \varGamma (x) =\varGamma ^{\prime}(x)/\varGamma (x)\), is known as the digamma function. Its derivatives ψ (n), n \( \geqslant \) 1, are known as the polygamma functions ψ n . These functions have the following representations [1, pp. 258–260] for x > 0 and each natural number n:
and
Several researchers have studied the properties of these functions. In 2007, refining the left inequality in [6, Theorem 4.8], Batir [45] obtained estimates for ψ n in terms of ψ or ψ k , with k < n. In particular, he proved, for x > 0 and \(n \in \mathbb{N}\):
and, for 1 \( \leqslant \) k \( \leqslant \) n − 1, x > 0,
He also proved, for example, the difference formula
where \(\alpha = {(n!\zeta (n + 1))}^{-1/n}\) and \(\beta = {((n - 1)!)}^{-1/n}\) are best possible, and the sharp estimates
where γ is the Euler-Mascheroni constant.
In 2010 Mortici [131] proved the following estimates, for x > 0 and n \( \geqslant \) 1, refining the work of Guo, Chen, and Qi [89]:
4.6 Completely Monotonic Functions
A function f is said to be completely monotonic on an interval I if (−1)n f (n)(x) \( \geqslant \) 0 for all x ∈ I and all nonnegative integers n. If this inequality is strict, then f is called strictly completely monotonic. Such functions occur in probability theory, numerical analysis, and other areas. Some of the most important completely monotonic functions are the gamma function and the digamma and polygamma functions. The Hausdorff-Bernstein-Widder theorem [189, Theorem 12b, p. 161] states that f is completely monotonic on [0, ∞) if and only if there is a nonnegative measure μ on [0, ∞) such that
for all x > 0. There is a well-written introduction to completely monotonic functions in [125].
In 2008 Batir [46] proved that the following function F a (x) related to the gamma function is completely monotonic on (0, ∞) if and only if a \( \geqslant \) 1∕4 and that − F a (x) is completely monotonic if and only if a \( \leqslant \) 0:
As a corollary he was able to prove, for x > 0, the inequality
with best constants \(\alpha = 1/4\) and β = 0, improving his earlier work with Alzer [9].
In 2010 Mortici [131] showed that for every n \( \geqslant \) 1, the functions \(f,g: (0,\infty ) \rightarrow \mathbb{R}\) given by
and
are completely monotonic on (0, ∞). As a corollary, since f(x) and g(x) are positive, he obtained estimates for | ψ n (x) | as finite series in negative powers of x.
Anderson and Qiu [13], as well as some other authors (see [2]), have studied the monotonicity properties of the function \(f(x) \equiv (\log \varGamma (x + 1))/x\). In 2011 Adell and Alzer [2] proved that f′ is completely monotonic on (−1, ∞).
In the course of pursuing research inspired by [13, 14] (see [53]), in 2012 Alzer [7] discussed properties of the function
which Qi and Guo [149] later conjectured to be completely monotonic on (0, ∞). In [53] Berg and Pedersen proved this conjecture.
In 2001 Berg and Pedersen [50] proved that the derivative of the function
is completely monotonic (see also [51]). This result extends the work of [13, 85]. Very recently, Berg and Pedersen [52] showed that the function
is a Pick function when a \( \geqslant \) 1, that is, it extends to a holomorphic function mapping the upper half plane into itself. The authors also considered the function
and proved that logf(x + 1) is a Stieltjes function and hence that f(x + 1) is completely monotonic on (0, ∞).
5 The Hypergeometric Function and Elliptic Integrals
The classical hypergeometric function is defined by
where \((a,n) \equiv a(a + 1)(a + 2)\cdots (a + n - 1)\) for \(n \in \mathbb{N}\) and (a, 0) = 1 for a ≠ 0. This function is so general that for proper choice of the parameters a, b, c, one obtains logarithms, trigonometric functions, inverse trigonometric functions, elliptic integrals, or polynomials of Chebyshev, Legendre, Gegenbauer, Jacobi, and so on (see [1, Chap. 15]).
5.1 Hypergeometric Functions
The Bernoulli inequality [126, p. 34] may be written as
where c > 1, t > 0. In [111] some Bernoulli-type inequalities have been obtained.
It is well known that in the zero-balanced case \(c = a + b\) the hypergeometric function F(a, b; c; x) has a logarithmic singularity at x = 1 (cf. [18, Theorem 1.19(6)]). Moreover, as a special case [1, 15.1.3],
Because of this connection, Vuorinen and his collaborators [110] have generalized versions of (32) to a wide class of hypergeometric functions. In the course of their investigation they have studied monotonicity and convexity/concavity properties of such functions. For example, for positive a, b let g(x) ≡ xF(a, b; a + b; x), x ∈ (0, 1). These authors have proved that \(G(x) \equiv \log g({e}^{x}/(1 + {e}^{x}))\) is concave on (−∞, ∞) if and only if \(1/a + 1/b \geqslant 1.\) And they have shown that, for fixed a, b ∈ (0, 1] and for x ∈ (0, 1), p > 0, the function
is increasing in p. In particular,
Motivated by the asymptotic behavior of F(x) = F(a, b; c; x) as x → 1−, Simić and Vuorinen have carried the above work further in [170], finding best possible bounds, when a, b, c > 0 and 0 < x, y < 1, for the quotient and difference
In 2009 Karp and Sitnik [108] obtained some inequalities and monotonicity of ratios for the generalized hypergeometric function. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function.
5.2 Complete Elliptic Integrals
For 0 < r < 1, the complete elliptic integrals of the first and second kind are defined as
and
respectively. Letting \(r^{\prime} \equiv \sqrt{1 - {r}^{2}}\), we often denote
These elliptic integrals have the hypergeometric series representations
5.3 The Landen Identities
The functions \(\mathcal{K}\) and \(\mathcal{E}\) satisfy the following identities due to Landen [64, 163.01, 164.02]:
Using Landen’s transformation formulas, we have the following identities [177, Lemma 2.8]: For r ∈ (0, 1), let \(t = (1 - r)/(1 + r)\). Then
Generalizing a Landen identity, Simić and Vuorinen [171] have determined the precise regions in the ab-plane for which a Landen inequality holds for zero-balanced hypergeometric functions. They proved that for all a, b > 0 with ab \( \leqslant \) 1∕4 the inequality
holds for r ∈ (0, 1), while for a, b > 0 with \(1/a + 1/b \leqslant 4,\) the following reversed inequality is true for each r ∈ (0, 1):
In the rest of the ab-plane neither of these inequalities holds for all r ∈ (0, 1). These authors have also obtained sharp bounds for the quotient
in certain regions of the ab-plane.
Some earlier results on Landen inequalities for hypergeometric functions can be found in [152]. Recently, Baricz obtained Landen-type inequalities for generalized Bessel functions [29, 37].
Inspired by an idea of Simić and Vuorinen [171], Wang, Chu, and Jiang [188] obtained some inequalities for zero-balanced hypergeometric functions which generalize Ramanujan’s cubic transformation formulas.
5.4 Legendre’s Relation and Generalizations
It is well known that the complete elliptic integrals satisfy the Legendre relation [64, 110.10]:
This relation has been generalized in various ways. Elliott [86] proved the identity
where
Elliott proved this formula by a clever change of variables in multiple integrals. Another proof, based on properties of the hypergeometric differential equation, was suggested without details in [25, p. 138], and the missing details were provided in [20]. It is easy to see that Elliott’s formula reduces to the Legendre relation when \(\lambda =\mu =\nu = 0\) and x = r 2.
Another generalization of the Legendre relation was given in [19]. With the notation
the authors considered the function
proving, in particular, that
This reduces to Elliott’s formula in case \(\lambda =\nu = 1/2 - a\) and \(\mu = c + a - 3/2\). In [19] it was conjectured that for a, b ∈ (0, 1), a + b ≤ 1( ≥ 1), \(\mathcal{L}(a,b,c,r)\) is concave (convex) as a function of r on (0, 1). In [107] Karatsuba and Vuorinen determined, in particular, the exact regions of abc-space in which the function \(\mathcal{L}(a,b,c,r)\) is concave, convex, constant, positive, negative, zero, and where it attains its unique extremum.
In [27] Balasubramanian, Naik, Ponnusamy, and Vuorinen obtained a differentiation formula for an expression involving hypergeometric series that implies Elliott’s identity. This paper contains a number of other significant results, including a proof that Elliott’s identity is equivalent to a formula of Ramanujan [54, p. 87, Entry 30] on the differentiation of quotients of hypergeometric functions.
5.5 Some Approximations for \(\mathcal{K}(r)\) by arthr
Anderson, Vamanamurthy, and Vuorinen [15] approximated \(\mathcal{K}(r)\) by the inverse hyperbolic tangent function arth, obtaining the inequalities
for 0 < r < 1. Alzer and Qiu [11] refined (37) as
with the best exponents 3∕4 and 1 for (arthr)∕r on the left and right, respectively. Seeking to improve the exponents in (38), they conjectured that the double inequality
holds for all 0 < r < 1, with best constants α = 0 and \(\beta = 1/4\). Very recently Chu et al. [81] gave a proof for this conjecture.
András and Baricz [24] presented some improved lower and upper bounds for \(\mathcal{K}(r)\) involving the Gaussian hypergeometric series.
5.6 Approximations for \(\mathcal{E}(r)\)
In [90] Guo and Qi have obtained new approximations for \(\mathcal{E}(r)\) as well as for \(\mathcal{K}(r)\). For example, they showed that, for 0 < r < 1,
In recent work [82, 178, 185] Chu et al. have obtained estimates for \(\mathcal{E}(r)\) in terms of rational functions of the arithmetic, geometric, and root-square mean, implying new inequalities for the perimeter of an ellipse.
5.7 Generalized Complete Elliptic Integrals
For 0 < a < min{c, 1} and 0 < b < c \( \leqslant \) a + b, define the generalized complete elliptic integrals of the first and second kind on [0, 1] by [95]
for r ∈ (0, 1), \(r^{\prime} = \sqrt{1 - {r}^{2}}\). The end values are defined by limits as r tends to 0+ and 1−, respectively. Thus,
and
Note that the restrictions on the parameters a, b, and c ensure that the function \(\mathcal{K}_{a,b,c}\) is increasing and unbounded, whereas \(\mathcal{E}_{a,b,c}\) is decreasing and bounded, as in the classical case \(a = b = 1/2\), c = 1.
Heikkala, Lindén, Vamanamurthy, and Vuorinen [94, 95] derived several differentiation formulas and obtained sharp monotonicity and convexity properties for certain combinations of the generalized elliptic integrals. They also constructed a conformal mapping sn a, b, c from a quadrilateral with internal angles bπ, (c − b)π, (1 − a)π, and \((1 - c + a)\pi\) onto the upper half plane. These results generalize the work of [19]. For some particular parameter triples (a, b, c), there are very recent results by many authors [37, 181, 206, 209].
With suitable restrictions on the parameters a, b, c, Neuman [135] has obtained bounds for \(\mathcal{K}_{a,b,c}\) and \(\mathcal{E}_{a,b,c}\) and for certain combinations and products of them. He has also proved that these generalized elliptic integrals are logarithmically convex as functions of the first parameter.
In 2007 Baricz [33, 36, 38] established some Turán-type inequalities for Gaussian hypergeometric functions and generalized complete elliptic integrals. He also studied the generalized convexity of the zero-balanced hypergeometric functions and generalized complete elliptic integrals [31] (see also [30, 32, 37]). Very recently, Kalmykov and Karp [103, 104] have studied log-convexity and log-concavity for series involving gamma functions and derived many known and new inequalities for the modified Bessel and Kummer and generalized hypergeometric functions and ratios of the Gauss hypergeometric functions. In particular, they improved and generalized Baricz’s Turán-type inequalities.
5.8 The Generalized Modular Function and Generalized Linear Distortion Function
Let a, b, c > 0 with a + b ≥ c. A generalized modular equation of order (or degree) p > 0 is
The generalized modulus is the decreasing homeomorphism μ a, b, c : (0, 1) → (0, ∞), defined by
The generalized modular equation (43) can be written as
With \(p = 1/K\), K > 0, the solution of (43) is then given by
Here \(\varphi _{K}^{a,b,c}\) is called the (a, b, c)-modular function with degree \(p = 1/K\) [19, 94, 95]. Clearly the following identities hold:
In [94], the authors generalized the functional inequalities for the modular functions and Grötzsch function μ proved in [19] to hold also for the generalized modular functions and generalized modulus in the case \(b = c - a\). For instance, for 0 < a < c \( \leqslant \) 1 and K > 1, the inequalities
hold for all u, t ∈ (0, 1), with equality if and only if u = t, and
For the special case of \(a = 1/2\) and c = 1 the readers are referred to [18]. Wang et al. [181] presented several sharp inequalities for the generalized modular functions with specific choice of parameters c = 1 and \(b = 1 - a\).
A linearization for the generalized modular function is also presented in [94] as follows: Let p: (0, 1) → (−∞, ∞) and q: (−∞, ∞) → (0, 1) be given by \(p(x) = 2\log (x/x^{\prime})\) and \(q(x) = {p}^{-1}(x) = \sqrt{{e}^{x } /({e}^{x } + 1)}\), respectively, and for a ∈ (0, 1), c ∈ (a, 1], K ∈ (1, ∞), let \(g,h: (-\infty,\infty ) \rightarrow (-\infty,\infty )\) be defined by \(g(x) = p(\varphi _{K}^{a,c-a,c}(q(x)))\) and \(h(x) = p(\varphi _{1/K}^{a,c-a,c}(q(x)))\). Then
In the same paper the authors also studied how these generalized functions depend on the parameter c. Corresponding results for the case c = 1 can be found in the articles [19, 153, 204].
Recently Bhayo and Vuorinen [55] have studied the Hölder continuity and submultiplicative properties of \(\varphi _{K}^{a,b,c}(r)\) in the case where c = 1 and \(b = 1 - a\) and have obtained several sharp inequalities for \(\varphi _{K}^{a,1-a,1}(r).\)
For x, K ∈ (0, ∞), define
and the generalized linear distortion function
For \(a = 1/2\), these two functions reduce to the well-known special case denoted by η K (x) and λ(K), respectively, which play a crucial role in quasiconformal theory. Several inequalities for these functions have been obtained as an application of the monotonicity and convexity of certain combinations of these functions and some elementary functions; see [55, 80, 122, 123, 180, 203]. For instance, the following chain of inequalities is proved in [80]: for a ∈ (0, 1∕2], K ∈ (1, ∞) and x, y ∈ (0, ∞),
with equality if and only if x = y.
6 Inequalities for Power Series
The following theorem [95, Theorem 4.3] is an interesting tool in simplified proofs for monotonicity of the quotient of two power series:
Theorem 6.1.
Let \(\sum _{n=0}^{\infty }a_{n}{x}^{n}\) and \(\sum _{n=0}^{\infty }b_{n}{x}^{n}\) be two real power series converging on the interval (−R,R). If the sequence {a n ∕b n } is increasing (decreasing) and b n > 0 for all n, then the function
is also increasing (decreasing) on (0,R). In fact, the function
has positive Maclaurin coefficients.
A more general version of this theorem appears in [58] and [147, Lemma 2.1]. This kind of rule also holds for the quotient of two polynomials instead of two power series (cf. [95, Theorem 4.4]):
Theorem 6.2.
Let \(f_{n}(x) =\sum _{ k=0}^{n}a_{k}{x}^{k}\) and \(g_{n}(x) =\sum _{ k=0}^{n}b_{k}{x}^{k}\) be two real polynomials, with b k > 0 for all k. If the sequence {a k ∕b k } is increasing (decreasing), then so is the function f n (x)∕g n (x) for all x > 0. In fact, \(g_{n}f_{n}^{\prime} - f_{n}g_{n}^{\prime}\) has positive (negative) coefficients.
In 1997 Ponnusamy and Vuorinen [147] refined Ramanujan’s work on asymptotic behavior of the hypergeometric function and also obtained many inequalities for the hypergeometric function by making use of Theorem 6.1. Many well-known results of monotonicity and inequalities for complete elliptic integrals have been extended to the generalized elliptic integrals in [94, 95].
Motivated by an open problem of Anderson et al. [16], in 2006 Baricz [30] considered ratios of general power series and obtained the following theorem. Note the similarity of the last inequality in Theorem 6.3 with the left-hand side of the inequality (45).
Theorem 6.3.
Suppose that the power series \(f(x) =\sum _{ n=0}^{\infty }a_{n}{x}^{n}\) with a n > 0 for all n \( \geqslant \) 0 is convergent for all x ∈ (0,1) and also that the sequence \(\{(n + 1)a_{n+1}/a_{n} - n\}_{n\geqslant 0}\) is strictly decreasing. Let the function m f : (0,1) → (0,∞) be defined as \(m_{f}(r) = f(1 - {r}^{2})/f({r}^{2})\) . Then
for all \(r_{1},r_{2},\ldots,r_{k} \in (0,1)\) , where equality holds if and only if \(r_{1} = r_{2} = \cdots = r_{k}\) . In particular, for k = 2 the inequalities
hold for all r 1 ,r 2 ∈ (0,1), and in all these inequalities equality holds if and only if r 1 = r 2.
The following Landen-type inequality for power series is also due to Baricz [29].
Theorem 6.4.
Suppose that the power series \(f(x) =\sum _{ n=0}^{\infty }a_{n}{x}^{n}\) with a n > 0 for all n \( \geqslant \) 0 is convergent for all x ∈ (0,1) and that for a given δ > 1 the sequence \(\{n!a_{n}/{(\log \delta )}^{n}\}_{n\geqslant 0}\) is decreasing. If λ f (x) = f(x 2 ), then
holds for all r ∈ (0,1) and \(\rho {\geqslant \delta }^{4\sqrt{2}-5}\).
Anderson, Vamanamurthy, and Vuorinen [22] studied generalized convexity and gave sufficient conditions for generalized convexity of functions defined by Maclaurin series. These results yield a class of new inequalities for power series which improve some earlier results obtained by Baricz. More inequalities for power series can be found in [37, 80].
In 1928 T. Kaluza gave a criterion for the signs of the power series of a function that is the reciprocal of another power series.
Theorem 6.5 ([105]).
Let \(f(x) =\sum _{n\geq 0}a_{n}{x}^{n}\) be a convergent Maclaurin series with radius of convergence r > 0. If a n > 0 for all n \( \geq \) 0 and the sequence \(\{a_{n}\}_{n\geq 0}\) is log-convex, that is, for all n \( \geq \) 0
then the coefficients b n of the reciprocal power series \(1/f(x) =\sum _{n\geq 0}b_{n}{x}^{n}\) have the following properties: \(b_{0} = 1/a_{0} > 0\) and b n \( \leqslant \) 0 for all n \( \geqslant \) 1.
In 2011 Baricz, Vesti, and Vuorinen [42] showed that the condition (48) cannot be replaced by the condition
for any t > 0. However, it is not known whether the condition (48) is necessary.
In 2009 Koumandos and Pedersen [115, Lemma 2.2] proved the following interesting result, which deals with the monotonicity properties of the quotient of two series of functions.
Theorem 6.6.
Suppose that a k > 0, b k > 0 and that {u k (x)} is a sequence of positive C 1 -functions such that the series
converge absolutely and uniformly over compact subsets of [0,∞). Define
-
1.
If the logarithmic derivatives u k ′(x)∕u k (x) form an increasing sequence of functions and if a k ∕b k decreases (resp. increases), then f(x) decreases (resp. increases) for x \( \geq \) 0.
-
2.
If the logarithmic derivatives u k ′(x)∕u k (x) form a decreasing sequence of functions and if a k ∕b k decreases (resp. increases), then f(x) increases (resp. decreases) for x \( \geq \) 0.
For inequalities of power series as complex functions, see [99–101] and the references therein.
7 Means
A homogeneous bivariate mean is defined as a continuous function \(\mathcal{M}: {\mathbb{R}}^{+} \times {\mathbb{R}}^{+} \rightarrow \mathbb{R}\) satisfying \(\min \{x,y\} \leq \mathcal{M}(x,y) \leq \max \{ x,y\}\) and \(\mathcal{M}(\lambda x,\lambda y) =\lambda \mathcal{M}(x,y)\) for all x, y, λ > 0. Important examples are the arithmetic mean A(a, b), the geometric mean G(a, b), the logarithmic mean L(a, b), the identric mean I(a, b), the root-square mean Q(a, b), and the power mean M r (a, b) of order r defined, respectively, by
7.1 Power Means
The weighted power means are defined by
\(M_{0}(\omega;a,b) \equiv {a}^{\omega }{b}^{1-\omega }\), with weights ω, 1 −ω > 0. The power means are the equally weighted means \(M_{\lambda }(a,b) = M_{\lambda }(1/2;a,b)\). As a special case, we have \(M_{0}(1/2;a,b) = G(a,b)\).
In [114] Kouba studied the ratio of differences of power means
finding sharp bounds for ρ(s, t, p; a, b) in various regions of stp-space with a, b positive and a ≠ b. This work extends the results of Alzer and Qiu [10], Trif [175], Kouba [113], Wu [193], and Wu and Debnath [196]. Kouba also extended the range of validity of the following inequality, due to Wu and Debnath [196]:
to the set of real numbers r, t, s, p satisfying the conditions 0 < s < t < r and \(0 < p \leqslant (4t + 2s)/3.\)
7.2 Toader Means
If \(p: {\mathbb{R}}^{+} \rightarrow {\mathbb{R}}^{+}\) is a strictly monotonic function, then define
where a, b are positive real numbers. The Toader mean [174] of a and b is defined as T(a, b; p, n) ≡ p −1(f(a, b; p, n)). It is easy to see that the Toader mean is symmetric. For special choices of p, let T q, n (a, b) = T(a, b; p, n) if p(x) = x q with q ≠ 0 and T 0, n (a, b) = T(a, b; p, n) if p(x) = logx. The means T q, n belong to a large family of means called the hypergeometric means, which have been studied by Carlson and others [62, 65, 68]. It is easy to see that T q, n is homogeneous. In particular, we have
Furthermore, the Toader means are related to the complete elliptic integrals: for a \( \geqslant \) b > 0,
In 1997 Qiu and Shen [151] proved that, for all a, b > 0 with a ≠ b,
This inequality had been conjectured by Vuorinen [176]. Alzer and Qiu [10] proved the following best possible power mean upper bound:
Very recently, Chu and his collaborators [78, 79, 83] obtained several bounds for T 1, 2 with respect to some combinations of various means.
7.3 Seiffert Means
The Seiffert means S 1 and S 2 are defined by
and
It is well known that
Sándor proved similar results for Seiffert means [160, 161]:
and
The inequalities (49) and (50) are special cases of more general results obtained by Neuman and Sándor [139, 140].
7.4 Extended Means
Let a, b ∈ (0, ∞) be distinct and \(s,t \in \mathbb{R}\setminus \{0\}\), s ≠ t. We define the extended mean [172] with parameters s and t by
and also
We see that all the classical means belong to the family of extended means. For example, E 2, 1 = A, E 0, 0 = G, \(E_{-1,-2} = H\), and E 1, 0 = L and, more generally, M λ = E 2λ, λ for \(\lambda \in \mathbb{R}\). The reader is referred to the survey [148] for many interesting results on the extended mean.
In 2002 Hästö [91] studied a certain monotonicity property of ratios of extended means and Seiffert means, which he called a strong inequality. These strong inequalities were shown to be related to the so-called relative metric [92, 93].
7.5 Means and the Circular and Hyperbolic Functions
It is easy to check the following identities:
One can get many inequalities by combining the above identities and inequalities between means. For example, combining (49) and (52), we have
where the second inequality is the well-known Cusa-Huygens inequality, and combining (50), (53), and (55), we have
More inequalities on mean values and trigonometric and hyperbolic functions can be found in [136, 163, 165, 200, 208] and references therein.
7.6 Means and Hypergeometric Functions
In 2005 Richards [158] obtained sharp power mean bounds for the hypergeometric function: Let 0 < a, b \( \leq \) 1 and c > max{ − a, b}. If c \( \geq \) max{1 − 2a, 2b}, then
if and only if \(\lambda \leq \frac{a+c} {1+c}\). If c \( \leq \) min{1 − 2a, 2b}, then
if and only if \(\mu \geqslant \frac{a+c} {1+c}\). These inequalities generalize earlier results proved by Carlson [66].
For hypergeometric functions of form \(F{(1/2 - s,1/2 + s;1;1 - {r}^{p})}^{q}\), Borwein et al. [61] exhibited explicitly iterations similar to the arithmetic-geometric mean. Barnard et al. [43] presented sharp bounds for hypergeometric analogs of the arithmetic-geometric mean as follows: For 0 < α \( \leqslant \) 1∕2 and p > 0,
if and only if λ \( \leq \) 0 and \(\mu \geq p(1-\alpha )/2\).
Some other inequalities involving hypergeometric functions and bivariate means can be found in the very recent survey [44].
For any two power means M λ and M μ , a function f is called M λ, μ -convex if it satisfies
Recently many authors have proved that the zero-balanced Gaussian hypergeometric function is M λ, λ -convex when λ ∈ {−1, 0, 1}. For details see [22, 26, 37, 80]. Baricz [31] generalized these results to the M λ, λ -convexity of zero-balanced Gaussian hypergeometric functions with respect to a power mean for λ ∈ [0, 1]. Zhang et al. [205] extended these results to the case of M λ, μ -convexity with respect to two power means: For all a, b > 0, λ ∈ (−∞, 1], and μ ∈ [0, ∞) the hypergeometric function F(a, b; a + b; r) is M λ, μ -convex on (0, 1).
The following interesting open problem is presented by Baricz [36]:
Open Problem. If m 1 and m 2 are bivariate means, then find conditions on a 1 ,a 2 > 0 and c > 0 for which the inequality
holds true for all r ∈ (0,1), where \(F_{a}(r) = F(a,c - a;c;r)\).
7.7 Means and Quasiconformal Analysis
Special functions have always played an important role in the distortion theory of quasiconformal mappings. Anderson, Vamanamurthy, and Vuorinen [18] have systematically investigated classical special functions and their extensive applications in the theory of conformal invariants and quasiconformal mappings. Some functional inequalities for special functions in quasiconformal mapping theory involve the arithmetic mean, geometric mean, or harmonic mean. For example, for the well-known Grötzsch ring function μ and the Hersch-Pfluger distortion function \(\varphi _{K}\), the following inequalities hold for all s, t ∈ (0, 1) with s ≠ t:
and
Recently, Wang, Zhang, and Chu [182, 183] have extended these inequalities as follows:
and
Some similar results for the generalized Grötzsch function, generalized modular function, and other special functions related to quasiconformal analysis can be found in [155, 179, 184, 186, 187].
8 Epilogue and a View Toward the Future
In earlier work we have listed many open problems. See especially [14, pp. 128–131] and [18, p. 478]. Many of these problems are still open. In Sects. 4, 6, and 7 above, we have also mentioned some open problems.
Finally, we wish to suggest some ideas for further research. In a frequently cited paper [119] Lindqvist introduced in 1995 the notion of generalized trigonometric functions such as sin p , and presently there is a large body of literature about this topic. For the case p = 2 the classical functions are obtained. In 2010, Biezuner et al. [59] developed a practical numerical method for computing values of sin p . Recently, Takeuchi [173] has gone a step further, introducing functions depending on two parameters p and q that reduce to the p-functions of Lindqvist when p = q. In [56, 57, 112] the authors have continued the study of this family of generalized functions and have suggested that many properties of classical functions have a counterpart in this more general setting. It would be natural to generalize the properties of trigonometric functions cited in this survey to the (p, q)-trigonometric functions of Takeuchi.
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Acknowledgements
The authors wish to thank Á. Baricz, C. Berg, E. A. Karatsuba, C. Mortici, E. Neuman, H. L. Pedersen, S. Ponnusamy, and G. Tee for careful reading of this paper and for many corrections and suggestions. The research of Matti Vuorinen was supported by the Academy of Finland, Project 2600066611. Xiaohui Zhang is indebted to the Finnish National Graduate School of Mathematics and its Applications for financial support.
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Anderson, G.D., Vuorinen, M., Zhang, X. (2014). Topics in Special Functions III. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_11
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